Reduced Fuzzy Differential Transform Method For Solving Two-dimensional Fuzzy Volterra Integral Equations

In practical problems there are a large number of ambiguities,and the absoluteness of the “one or the other” of the classic set theory is limited in describing these ambi-guities,which promotes the emergence of fuzzy mathematics.As an important branch of fuzzy mathematics,fuzzy integral equations are widely used in fields such as fuzzy control,fuzzy economy,etc.At present,a large number of scholars have studied the one-dimensional fuzzy integral equation and obtained many effective numerical methods.However,there are few studies on the two-dimensional fuzzy integral equation.There-fore,in this thesis we propose a reduced fuzzy differential transform method for solving two-dimensional fuzzy Volterra integral equations.It is analogous to the research of fuzzy differential transform method and is a generalization of the reduced differential transform method.Firstly,on the basis of the fuzzy differential transform method,we give the definition of the reduced fuzzy differential transform and its inverse transform.The related prop-erties of the reduced fuzzy differential transform method are proved.That is,when the original function is in different functional forms,the specific solution formula of the re-duced fuzzy differential transform is given.And those properties are the main theoretical basis for the subsequent reduced fuzzy differential transform.Secondly,two generalized Gronwall inequalities are proved.One is the generalized two-dimensional Gronwall inequality.It is used to prove that the solution of the two-dimensional fuzzy Volterra integral equation exists and is unique when the kernel function satisfies the Lipschitz condition.This proof is different from the conventional fixed point principle method and successive approximation method,which reduces the restriction on the equation.The second is the generalized iterative Gronwall inequality,which is used for subsequent numerical stability analysis.Thirdly,we give a numerical method to solve the two-dimensional fuzzy Volterra integral equation by the reduced fuzzy differential transform method.First,the fuzzy integral equation is transformed into a system of integral equations by the r level set,then using the reduced fuzzy differential transform method the iterative equations can be obtained.After solving this iterative process,we can get coefficient function,and then get the approximate solution and even the exact solution of the equation.According to the generalized iterative Gronwall inequality,it is proved that numerical algorithm is stable in selecting the initial iteration value,and the error estimate of the method is given.Finally,linear and nonlinear numerical examples show that the reduced fuzzy differ-ential transform method is simple,effective and can converge quickly.Compared with the two-dimensional fuzzy differential transform method,our method has fewer iterative steps,requires less calculation and can obtain a high-precision approximate solution even the exact solution.